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A061914 Let H_n = n-th Hilbert matrix; sequence gives 1 / ( det(H_n) * denominator(permanent(H_n)) ).

%I A061914 #30 Jul 01 2025 16:05:30
%S A061914 1,1,1,27,567,1,1,1,7,9,5103,1275989841,992436543,48629390607,
%T A061914 169706648853,40257567,63,1,7,31,1,3969,25865973,117649,117649,16807,
%U A061914 49,9,81,117369,59049,33480783,930196594089,4238886345135097131,169560200598623521407
%N A061914 Let H_n = n-th Hilbert matrix; sequence gives 1 / ( det(H_n) * denominator(permanent(H_n)) ).
%H A061914 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Permanent.html">Permanent.</a>
%F A061914 a(n) = 1/(denominator(permanent(hilbert(n)))*det(hilbert(n))), where hilbert(n) denotes the n-th Hilbert matrix.
%p A061914 with(linalg): seq(1/(denom(permanent(hilbert(n)))*det(hilbert(n))), n=1..16);
%t A061914 Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]; f[n_] := Block[{i = Table[1/(i + j - 1), {i, n}, {j, n}]}, 1/(Det[i]Denominator[Permanent[i]])]; Table[ f[n], {n, 1, 18}] (* _Robert G. Wilson v_, Feb 06 2004 *)
%o A061914 (PARI) permRWN(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); n1=n-1; sg=1; m=1; nc=0; in=vector(n); x=in; for(i=1,n,x[i]=a[i,n]-sum(j=1,n,a[i,j])/2); p=prod(i=1,n,x[i]); while(m,sg=-sg; j=1; if((nc%2)!=0,j++; while(in[j-1]==0,j++)); in[j]=1-in[j]; nc+=2*in[j]-1; m=nc!=in[n1]; z=2*in[j]-1; for(i=1,n,x[i]+=z*a[i,j]); p+=sg*prod(i=1,n,x[i])); return(2*(2*(n%2)-1)*p)
%o A061914 for(n=1,23,a=mathilbert(n); print1(1/(matdet(a)*denominator(permRWN(a)))", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007
%o A061914 (PARI) for(n=1, 25, a=mathilbert(n); print1(1 / (matdet(a) * denominator(matpermanent(a)))", ")) \\ _Vaclav Kotesovec_, Aug 13 2021
%Y A061914 Cf. A005249.
%K A061914 nonn
%O A061914 1,4
%A A061914 _Asher Auel_, May 20 2001
%E A061914 a(18)-a(20) from _Robert G. Wilson v_, Feb 09 2004
%E A061914 a(21) from _Eric W. Weisstein_, Feb 19 2004
%E A061914 a(22) and a(23) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007
%E A061914 a(24)-a(34) from _Vaclav Kotesovec_, Aug 14 2021
%E A061914 a(35) from _Vaclav Kotesovec_, Aug 16 2021